An Existence Result for Nonlinear Fractional Differential Equations on Banach Spaces

  • Mouffak Benchohra1Email author,

    Affiliated with

    • Alberto Cabada2 and

      Affiliated with

      • Djamila Seba3

        Affiliated with

        Boundary Value Problems20092009:628916

        DOI: 10.1155/2009/628916

        Received: 30 January 2009

        Accepted: 15 May 2009

        Published: 22 June 2009

        Abstract

        The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of noncompactness.

        1. Introduction

        The theory of fractional differential equations has been emerging as an important area of investigation in recent years. Let us mention that this theory has many applications in describing numerous events and problems of the real world. For example, fractional differential equations are often applicable in engineering, physics, chemistry, and biology. See Hilfer[1], Glockle and Nonnenmacher [2], Metzler et al. [3], Podlubny [4], Gaul et al. [5], among others. Fractionaldifferential equations are also often an object of mathematical investigations; see the papers of Agarwal et al. [6], Ahmad and Nieto [7], Ahmad and Otero-Espinar [8], Belarbi et al. [9], Belmekki et al [10], Benchohra et al. [1113], Chang and Nieto [14], Daftardar-Gejji and Bhalekar [15], Figueiredo Camargo et al. [16], and the monographs of Kilbas et al. [17] and Podlubny [4].

        Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq1_HTML.gif , and so forth. the same requirements of boundary conditions. Caputo's fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville and Caputo types, see [18, 19].

        In this paper we investigate the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral conditions of the form
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ1_HTML.gif
        (11)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq2_HTML.gif is the Caputo fractional derivative, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq4_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq5_HTML.gif are given functions satisfying some assumptions that will be specified later, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq6_HTML.gif is a Banach space with norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq7_HTML.gif .

        Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary conditions are often encountered in various applications; it is worthwhile mentioning the applications of those conditions in the study of population dynamics [20] and cellular systems [21].

        Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors such as, for instance, Arara and Benchohra [22], Benchohra et al. [23, 24], Infante [25], Peciulyte et al. [26], and the references therein.

        In our investigation we apply the method associated with the technique of measures of noncompactness and the fixed point theorem of Mönch type. This technique was mainly initiated in the monograph of Bana and Goebel [27] and subsequently developed and used in many papers; see, for example, Bana and Sadarangoni [28], Guo et al. [29], Lakshmikantham and Leela [30], Mönch [31], and Szufla [32].

        2. Preliminaries

        In this section, we present some definitions and auxiliary results which will be needed in the sequel.

        Denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq8_HTML.gif the Banach space of continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq9_HTML.gif , with the usual supremum norm
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ2_HTML.gif
        (21)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq10_HTML.gif be the Banach space of measurable functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq11_HTML.gif which are Bochner integrable, equipped with the norm
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ3_HTML.gif
        (22)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq12_HTML.gif be the Banachspace of measurable functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq13_HTML.gif which are bounded, equipped with the norm
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ4_HTML.gif
        (23)

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq14_HTML.gif be the space of functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq15_HTML.gif , whose first derivative is absolutely continuous.

        Moreover, for a given set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq16_HTML.gif of functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq17_HTML.gif let us denote by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ5_HTML.gif
        (24)

        Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.

        Definition 2.1 (see [27]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq18_HTML.gif be a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq19_HTML.gif the bounded subsets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq20_HTML.gif . The Kuratowski measure of noncompactness is the map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq21_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ6_HTML.gif
        (25)

        Properties

        The Kuratowski measure of noncompactness satisfies some properties (for more details see [27]).

        (a) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq22_HTML.gif is compact ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq23_HTML.gif is relatively compact).

        (b) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq24_HTML.gif .

        (c) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq25_HTML.gif .

        (d) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq26_HTML.gif

        (e) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq27_HTML.gif

        (f) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq28_HTML.gif .

        Here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq30_HTML.gif denote the closure and the convex hull of the bounded set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq31_HTML.gif , respectively.

        For completeness we recall the definition of Caputo derivative of fractional order.

        Definition 2.2 (see [17]).

        The fractional order integral of the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq32_HTML.gif of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq33_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ7_HTML.gif
        (26)
        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq34_HTML.gif is the gamma function. When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq35_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq36_HTML.gif where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ8_HTML.gif
        (27)

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq37_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq38_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq39_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq40_HTML.gif .

        Here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq41_HTML.gif is the delta function.

        Definition 2.3 (see [17]).

        For a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq42_HTML.gif given on the interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq43_HTML.gif , the Caputo fractional-order derivative of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq44_HTML.gif , of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq45_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ9_HTML.gif
        (28)

        Here http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq47_HTML.gif denotes the integer part of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq48_HTML.gif .

        Definition 2.4.

        A map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq49_HTML.gif is said to be Carathéodory if

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq50_HTML.gif is measurable for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq51_HTML.gif

        (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq52_HTML.gif is continuous for almost each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq53_HTML.gif

        For our purpose we will only need the following fixed point theorem and the important Lemma.

        Theorem 2.5 (see [31, 33]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq54_HTML.gif be a bounded, closed and convex subset of a Banach space such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq55_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq56_HTML.gif be a continuous mapping of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq57_HTML.gif into itself. If the implication
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ10_HTML.gif
        (29)

        holds for every subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq58_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq59_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq60_HTML.gif has a fixed point.

        Lemma 2.6 (see [32]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq61_HTML.gif be a bounded, closed, and convex subset of the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq62_HTML.gif , G a continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq63_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq64_HTML.gif satisfies the Carathéodory conditions, and there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq65_HTML.gif such that for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq66_HTML.gif and each bounded set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq67_HTML.gif one has
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ11_HTML.gif
        (210)
        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq68_HTML.gif is an equicontinuous subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq69_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ12_HTML.gif
        (211)

        3. Existence of Solutions

        Let us start by defining what we mean by a solution of the problem (1.1).

        Definition 3.1.

        A function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq70_HTML.gif is said to be a solution of (1.1) if it satisfies (1.1).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq71_HTML.gif be continuous functions and consider the linear boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ13_HTML.gif
        (31)

        Lemma 3.2 (see [11]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq72_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq73_HTML.gif be continuous. A function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq74_HTML.gif is a solution of the fractional integral equation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ14_HTML.gif
        (32)
        with
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ15_HTML.gif
        (33)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ16_HTML.gif
        (34)

        if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq75_HTML.gif is a solution of the fractional boundary value problem (3.1).

        Remark 3.3.

        It is clear that the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq76_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq77_HTML.gif , and hence is bounded. Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ17_HTML.gif
        (35)

        For the forthcoming analysis, we introduce the following assumptions

        (H1)The functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq78_HTML.gif satisfy the Carathéodory conditions.

        (H2)There exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq79_HTML.gif , such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ18_HTML.gif
        (36)
        (H3)For almost each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq80_HTML.gif and each bounded set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq81_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ19_HTML.gif
        (37)

        Theorem 3.4.

        Assume that assumptions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq82_HTML.gif hold. If
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ20_HTML.gif
        (38)

        then the boundary value problem (1.1) has at least one solution.

        Proof.

        We transform the problem (1.1) into a fixed point problem by defining an operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq83_HTML.gif as
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ21_HTML.gif
        (39)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ22_HTML.gif
        (310)
        and the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq84_HTML.gif is given by (3.4). Clearly, the fixed points of the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq85_HTML.gif are solution of the problem (1.1). Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq86_HTML.gif and consider the set
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ23_HTML.gif
        (311)

        Clearly, the subset http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq87_HTML.gif is closed, bounded, and convex. We will show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq88_HTML.gif satisfies the assumptions of Theorem 2.5. The proof will be given in three steps.

        Step 1.

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq89_HTML.gif is continuous.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq90_HTML.gif be a sequence such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq91_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq92_HTML.gif . Then, for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq93_HTML.gif ,

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ24_HTML.gif
        (312)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq94_HTML.gif be such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ25_HTML.gif
        (313)
        By (H2) we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ26_HTML.gif
        (314)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq96_HTML.gif are Carathéodory functions, the Lebesgue dominated convergence theorem implies that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ27_HTML.gif
        (315)

        Step 2.

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq97_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq98_HTML.gif into itself.

        For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq99_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq100_HTML.gif and (3.8) we have for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq101_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ28_HTML.gif
        (316)

        Step 3.

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq102_HTML.gif is bounded and equicontinuous.

        By Step 2, it is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq103_HTML.gif is bounded.

        For the equicontinuity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq104_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq105_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq107_HTML.gif . Then

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ29_HTML.gif
        (317)

        As http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq108_HTML.gif , the right-hand side of the above inequality tends to zero.

        Now let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq109_HTML.gif be a subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq110_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq111_HTML.gif .

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq112_HTML.gif is bounded and equicontinuous, and therefore the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq113_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq114_HTML.gif . By (H3), Lemma 2.6, and the properties of the measure http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq115_HTML.gif we have for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq116_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ30_HTML.gif
        (318)
        This means that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ31_HTML.gif
        (319)

        By (3.8) it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq117_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq118_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq119_HTML.gif , and then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq120_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq121_HTML.gif . In view of the Ascoli-Arzelà theorem, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq122_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq123_HTML.gif . Applying now Theorem 2.5 we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq124_HTML.gif has a fixed point which is a solution of the problem (1.1).

        4. An Example

        In this section we give an example to illustrate the usefulness of our main results. Let us consider the following fractional boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ32_HTML.gif
        (41)
        Set
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ33_HTML.gif
        (42)

        Clearly, conditions (H1),(H2) hold with

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ34_HTML.gif
        (43)
        From (3.4) the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq125_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ35_HTML.gif
        (44)
        From (4.4), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ36_HTML.gif
        (45)
        A simple computation gives
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ37_HTML.gif
        (46)
        Condition (3.8) is satisfied with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq126_HTML.gif . Indeed
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_Equ38_HTML.gif
        (47)

        which is satisfied for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq127_HTML.gif . Then by Theorem 3.4 the problem (4.1) has a solution on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F628916/MediaObjects/13661_2009_Article_866_IEq128_HTML.gif .

        Declarations

        Acknowledgments

        The authors thank the referees for their remarks. The research of A. Cabada has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.

        Authors’ Affiliations

        (1)
        Laboratoire de Mathématiques, Université de Sidi Bel-Abbès
        (2)
        Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela
        (3)
        Département de Mathématiques, Université de Boumerdès

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